BINARY INTEGERS
A binary integer x is a finite sequence of the digits 0
and 1, which we write symbolically as
x = (amam−1 · · · a2a1a0)2
where I insert the parentheses with subscript ()2 in
order to make clear that the number is binary. The
above has the decimal equivalent
x = am2m + am−12m−1 + · · · + a121 + a0
For example, the binary integer x = (110101)2 has
the decimal value
x = 25 + 24 + 22 + 20 = 53
The binary integer x = (111 · · · 1)2 with m ones has
the decimal value
x = 2m−1 + · · · + 21 + 1 = 2m − 1
DECIMAL TO BINARY INTEGER CONVERSION
Given a decimal integer x we write
x = (amam−1 · · · a2a1a0)2
= am2m + am−12m−1 + · · · + a121 + a0
Divide x by 2, calling the quotient x1. The remainder
is a0, and
x1 = am2m−1 + am−12m−2 + · · · + a120
Continue the process. Divide x1 by 2, calling the quotient x2. The remainder is a1, and
x2 = am2m−2 + am−12m−3 + · · · + a220
After a finite number of such steps, we will obtain all
of the coefficients ai, and the final quotient will be
zero.
Try this with a few decimal integers.
EXAMPLE
The following shortened form of the above method is
convenient for hand computation. Convert (11)10 to
binary.
√
b2 11c = √
5
= x1
a0 = 1
b2 5c = √
2
= x2
a1 = 1
b2 2c = √
1
= x3
a2 = 0
b2 1c = 0 = x4
a3 = 1
In this, the notation bbc denotes the largest integer
√
≤ b, and the notation 2 n denotes the quotient resulting from dividing 2 into n. From the above calculation, (11)10 = (1011)2.
BINARY FRACTIONS
A binary fraction x is a sequence (possibly infinite) of
the digits 0 and 1:
x = (.a1a2a3 · · · am · · · )2
= a12−1 + a22−2 + a32−3 + · · ·
For example, x = (.1101)2 has the decimal value
x = 2−1 + 2−2 + 2−4
= .5 + .25 + .0625 = 0.8125
Recall the formula for the geometric series
n
X
1 − rn+1
i
r =
,
1−r
i=0
r 6= 1
Letting n → ∞ with |r| < 1, we obtain the formula
∞
X
i=0
ri =
1
,
1−r
|r| < 1
Using this,
(.0101010101010 · · · )2 = 2−2 + 2−4 + 2−6 + · · ·
³
´
−2
−2
−4
1 + 2 + 2 + ···
= 2
which sums to the fraction 1/3.
Also,
(.11001100110011 · · · )2
= 2−1 + 2−2 + 2−5 + 2−6 + · · ·
8.
and this sums to the decimal fraction 0.8 = 10
DECIMAL TO BINARY FRACTION CONVERSION
In
x1 = (.a1a2a3 · · · am · · · )2
= a12−1 + a22−2 + a32−3 + · · ·
we multiply by 2. The integer part will be a1; and
after it is removed we have the binary fraction
x2 = (.a2a3 · · · am · · · )2
= a22−1 + a32−2 + a42−3 + · · ·
Again multiply by 2, obtaining a2 as the integer part of
2x2. After removing a2, let x3 denote the remaining
number. Continue this process as far as needed.
For example, with x = 15 , we have
x1 = .2; 2x1 = .4; x2 = .4 and a1 = 0
2x2 = .8; x3 = .8 and a2 = 0
2x3 = 1.6; x4 = .6 and a2 = 1
Continue this to get the pattern
(.2)10 = (.00110011001100 · · · )2
ADDITION TABLE
+
1
10
11
100
101
1
10
11
100
101
110
11 100
101
110
111
10
110
111 1000
11 100 101
111 1000 1001
100 101 110
101 110 111 1000 1001 1010
MULTIPLICATION TABLE
×
1
10
11
100
101
1
1
10
11
100
101
10
100
110
1000
1010
10
11
110 1001
1100
1111
11
100 100 1000 1100 10000 10100
101 101 1010 1111 10100 11001